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Line equation containing a point; did I get this right

We have the vectors:
a= (1,0,1)

b= (0,1,-1)

u=(-sqrt3,0,1)

v=(1,0,-sqrt3)

a and b are position vectors.

Line 1 contains a and is parallel to u (so direction vector for Line 1 is constant times u)
Line 2 contains b and is parallel to v (so direction vector for Line 1 is constant times v)


Question: Does Line 1 and Line 2 intersect?

I'm guessing ''contains'' means what it says i.e position vector a is on Line 1 and position vector b is on Line 2

(Instead of a Greek letter like lambda for the constant I am using ''J'' here randomly)

I got for Line 1

x=1-sqrt3J
y=0
z=1+J

and for Line 2

x=J
y=1
z=-1-sqrt3J

Equating both we see that y=0=1 which cannot hold therefore the lines do not intersect

Something told me I had to check this
I think you are correct.

r=i+k+λ(3i+k) \mathbf{r} = \mathbf{i} + \mathbf{k} + \lambda(\sqrt{3} \mathbf{i} + \mathbf{k})
s=j−k+μ(i−3k) \mathbf{s} = \mathbf{j} - \mathbf{k} + \mu(\mathbf{i} - \sqrt{3}\mathbf{k})

(1−3λ)i+(1+λ)k=μi+j+(−1−μ3)k(1 - \sqrt{3}\lambda)\mathbf{i} + (1 + \lambda)\mathbf{k} = \mu \mathbf{i} + \mathbf{j} + (-1-\mu \sqrt{3})\mathbf{k}

Equating coefficients of j\mathbf{j} and you get 0=10 = 1, so they don't intersect.
(edited 9 years ago)
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MSI_10
OP
15
Original post by GingerCodeMan
I think you are correct.

r=i+k+λ(3i+k) \mathbf{r} = \mathbf{i} + \mathbf{k} + \lambda(\sqrt{3} \mathbf{i} + \mathbf{k})
s=j−k+μ(i−3k) \mathbf{s} = \mathbf{j} - \mathbf{k} + \mu(\mathbf{i} - \sqrt{3}\mathbf{k})

(1−3i)i+(1+λ)k)=μi+j+(−1−μ3)k(1 - \sqrt{3}\mathbf{i})\mathbf{i} + (1 + \lambda)\mathbf{k}) = \mu \mathbf{i} + \mathbf{j} + (-1-\mu \sqrt{3})\mathbf{k}

Equating coefficients of j\mathbf{j} and you get 0=10 = 1, so they don't intersect.


Thank you +1rep

For clarification, here is the question. It is part iv

Vector q.png



For part V I got (-1-sqrt3/2, 1, 1/2+sqrt3)

and for (VI).. where should I begin? I'm a little bit confused when it says the plane it contains L1 i.e I know it has
x=1-sqrt3J
y=0
z=1+K

but what should I do with this parametric?
Sorry, I've never planes. I could have a go, but my answer probably wouldn't be very reliable :P

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